Linearized stability of traveling cell solutions arising from a moving boundary problem

In 2003, Mogilner and Verzi proposed a one-dimensional model on the crawling movement of a nematode sperm cell. Under certain conditions, the model can be reduced to a moving boundary problem for a single equation involving the length density of the bundled filaments inside the cell. It follows from the results of Choi, Lee and Lui (2004) that this simpler model possesses traveling cell solutions. In this paper, we show that the spectrum of the linear operator, obtained from linearizing the evolution equation about the traveling cell solution, consists only of eigenvalues and there exists $\mu > 0$ such that if $\lambda$ is a real eigenvalue, then $\lambda \leq -\mu$. We also provide strong numerical evidence that this operator has no complex eigenvalue.